Decimal numbers are any usual fraction, which can be written as a fraction with the number \(10^{n}\) (n∈N) in the denominator. *For example,* \(\frac{5}{100} , \frac{236}{1000} , \frac{389}{100}\) . Decimal numbers are written by using dot or comma. An \(\color{blue} {integer}\) is located before a decimal point, and the \(\color{green} {decimal}\) is located after the decimal point. *For example,* \( \frac{5}{100} = 0.05\) ; \( \frac{236}{1000} = 0.236\) ; \( \frac{389}{100} = 3,89\). We can write as many zeroes before the numbers that are located before a decimal point and after numbers that are located after decimal point as you want, and our number value will not change. *For example, *00005,750000=5,75; 0,63000=0,63; 0005,76=5,76.

After the decimal point, the quantity of numbers should be equal to the quantity of zeroes in the denominator. *For example, \( \frac{5}{100} =0,05 ; \frac{236}{1000} = 0,236\) .*

Before the decimal point, the following ranks are located: units of tens, hundreds and other. After the decimal point is located: ranks of tenth, hundredth, thousandth and other.

If we can’t convert the denominator of the fraction into a number of \(10^{n}\) , then we can divide the numerator into a denominator for getting a decimal number. *For example, \( \frac{35}{94} ≈ 0,37234\)*

*Kinds of decimal numbers*

* Exact*There are decimal numbers, which can be written fully. They contain a finite quantity of numbers. These numbers can be easily converted back into a fraction.

*For example:*0,357

This number has three numbers after the decimal point and can be easily converted into a fraction: \( \frac{357}{1000}\)

* Recurring*There are decimal numbers which have an infinite quantity of numbers, which periodically repeat. They can be written like a usual decimal point, but usually the part that repeats is written in brackets. Recurring decimal number, which hasn’t got other numbers between the decimal point and \( \color{green} {period}\) , is called a

__net recurring decimal number__. If, between the decimal point and the period the other number is located, this decimal is called a

__mixed recurring decimal number__.

*For example :*1,37373737… or 1,\(\color{green} ({37})\) is a net recurring decimal number.

3,5626262… or 3,5 \(\color{green} ({62})\) is a mixed recurring decimal number

* Other decimals*There are infinite non-recurring decimal numbers. They are irrational numbers. These are numbers which can’t be written as a usual fraction.

*For example:*\( \pi =3,14159… ; \sqrt{2} = 1,4142135…\)

*Rounding off the decimal numbers*

In order to round off the decimal number to the definite rank after a comma, we should check what the number in the following rank is. If this number is \( \color{blue} {5} \) (or bigger than \( \color{blue} {5}\) ), then we should add 1 to the previous number. If this number is smaller than \( \color{blue} {5}\) , then we should write the previous number without changes.

*For example: *round off 2,547 to the hundredth

Number 6 denotes the hundredth and 7 > 5, that way we add 1 to 4

\(2,547\approx 2,55\) .

*Adding of decimal numbers *

In order to add two decimal numbers, we should add corresponding ranks to each other. We should write thousandth under thousandth, hundredth under hundredth and all ranks under corresponding ranks.

*For example: \( \color{green} {\color{blue} {\color{red} {\color{green} {\color{blue} {2},3}7} + 1},6} \)*

\( \frac{+2,371,60}{=3,97}\)

*Subtraction of decimal numbers *

To subtract a decimal number from another, we should subtract corresponding ranks from each other. We should write thousandth under thousandth, hundredth under hundredth and all ranks under corresponding ranks.

*For example:* *\( \color{green} {\color{blue} {\color{red} {\color{green} {\color{blue} {2},3}7} – 1},6} \)*

\( \frac{-2,371,60}{=0,77}\)

*Multiplying of decimal numbers*

To multiply a decimal number by \(10^{n}\) , we need move the decimal point to n numbers to the right.

*For example:
*

To divide a decimal number into an integer, we need to ignore the comma and multiply decimal numbers like integers. Then we need to set apart quantities of numbers, equal to quantities of numbers in the divisor.

To multiply two decimal numbers, we need multiply them as usual numbers and then separate the product with the help of a decimal point. After the point, it should be quantities of numbers equal to the sum of the numbers after the decimal point in the first and second numbers.

*For example:* 2,37∙1,6

*Division of decimal numbers*

To divide a decimal number into \(10^{n}\) , we need to move the decimal point to n numbers to the left.

To divide decimal number into integer, we need to ignore the comma and divide decimal numbers like integers. Then, we need to set it apart in the fraction, a result of division of the integer part.

To divide one decimal number into other, first we need to multiply these two numbers by \(10^{n}\) , where n is the quantity of numbers after the decimal point in the denominator. Then, we can divide this decimal number like usual numbers.

*For example: \( 2,4 \div 1,6\)
Multiply 2,4 and 1,6 by \(10^{1}\):
\( 24 \div 16 \)
We can then ignore the comma and divide, like an integer:
\( 24 \div 16 = 1,5 \)
The result of division of the integer part is 1
\( 2,4 \div 1,6 = 1,5\) .*